mohammadloo
2007/5/01, 08:23 AM
Simple Pendulum
A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant (http://hyperphysics.phy-astr.gsu.edu/hbase/sound/reson.html#resdef) system with a single resonant frequency. For small amplitudes, the period (http://hyperphysics.phy-astr.gsu.edu/hbase/sound/sound.html#c2)of such a pendulum can be approximated by:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c2)function cper(){fh=document.forms[0];def();fh.t.value=2*Math.PI*Math.sqrt(fh.L.value*. 01/fh.g.value)}function clen(){fh=document.forms[0];def();fh.L.value=100*fh.t.value*fh.t.value*fh.g.v alue/(4*Math.PI*Math.PI)}function cg(){fh=document.forms[0];def();fh.g.value=fh.L.value*.01*(4*Math.PI*Math.P I)/(fh.t.value*fh.t.value)}function ul(l){fh=document.forms[0];fh.L.value=l;fh.Lm.value=l/100}function def(){fh=document.forms[0];if (fh.L.value==0)fh.L.value=25;if (fh.g.value==0)fh.g.value=9.8;if (fh.t.value==0)fh.t.value=1}For pendulum length (http://javascript<b></b>:var cal=clen())
L = cm = m and acceleration of gravity (http://javascript<b></b>:var cal=cg())
g = m/s2the pendulum period (http://javascript<b></b>:var cal=cper()) is
T = s(Enter data for two of the variables and then click on the active text for the third variable to calculate it.)
This expression for period is reasonably accurate for angles of a few degrees, but the treatment of the large amplitude pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pendl.html#c1) is much more complex. http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gifIf the rod is not of negligible mass, then it must be treated as a physical pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pendp.html#c1).
Pendulum Motion
The motion of a simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1) is like simple harmonic motion (http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1) in that the equation for the angular displacement is
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend3.gifShow (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3)which is the same form as the motion of a mass on a spring:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend4.gifThe anglular frequency of the motion is then given by http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend5.gifcompared tohttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend6.giffor a mass on a spring. http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gif The frequency of the pendulum in Hz is given byhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend7.gifand the period of motion is then http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend8.gifPeriod of Simple Pendulum
A point mass hanging on a massless string is an idealized example of a simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1). When displaced from its equilibrium (http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html#equi) point, the restoring force which brings it back to the center is given by:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend9.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c4)For small angles θ, we can use the approximation
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend10.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/trgser.html#c1)in which case Newton's 2nd law (http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#fma) takes the form
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend11.gifEven in this approximate case, the solution of the equation uses calculus and differential equations. The differential equation (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c5) is
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend12.gifand for small angles θ the solution is:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend13.gifhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend14.gifPendulum Equation
The equation of motion for the simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1) for sufficiently small amplitude has the form
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend16.gifwhich when put in angular form (http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#rq) becomes
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend17.gifThis differential equation (http://hyperphysics.phy-astr.gsu.edu/hbase/diff2.html#c2) is like that for the simple harmonic oscillator (http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c2) and has the solution:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend13.gifhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gif
A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant (http://hyperphysics.phy-astr.gsu.edu/hbase/sound/reson.html#resdef) system with a single resonant frequency. For small amplitudes, the period (http://hyperphysics.phy-astr.gsu.edu/hbase/sound/sound.html#c2)of such a pendulum can be approximated by:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c2)function cper(){fh=document.forms[0];def();fh.t.value=2*Math.PI*Math.sqrt(fh.L.value*. 01/fh.g.value)}function clen(){fh=document.forms[0];def();fh.L.value=100*fh.t.value*fh.t.value*fh.g.v alue/(4*Math.PI*Math.PI)}function cg(){fh=document.forms[0];def();fh.g.value=fh.L.value*.01*(4*Math.PI*Math.P I)/(fh.t.value*fh.t.value)}function ul(l){fh=document.forms[0];fh.L.value=l;fh.Lm.value=l/100}function def(){fh=document.forms[0];if (fh.L.value==0)fh.L.value=25;if (fh.g.value==0)fh.g.value=9.8;if (fh.t.value==0)fh.t.value=1}For pendulum length (http://javascript<b></b>:var cal=clen())
L = cm = m and acceleration of gravity (http://javascript<b></b>:var cal=cg())
g = m/s2the pendulum period (http://javascript<b></b>:var cal=cper()) is
T = s(Enter data for two of the variables and then click on the active text for the third variable to calculate it.)
This expression for period is reasonably accurate for angles of a few degrees, but the treatment of the large amplitude pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pendl.html#c1) is much more complex. http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gifIf the rod is not of negligible mass, then it must be treated as a physical pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pendp.html#c1).
Pendulum Motion
The motion of a simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1) is like simple harmonic motion (http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1) in that the equation for the angular displacement is
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend3.gifShow (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3)which is the same form as the motion of a mass on a spring:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend4.gifThe anglular frequency of the motion is then given by http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend5.gifcompared tohttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend6.giffor a mass on a spring. http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gif The frequency of the pendulum in Hz is given byhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend7.gifand the period of motion is then http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend8.gifPeriod of Simple Pendulum
A point mass hanging on a massless string is an idealized example of a simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1). When displaced from its equilibrium (http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html#equi) point, the restoring force which brings it back to the center is given by:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend9.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c4)For small angles θ, we can use the approximation
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend10.gif Show (http://hyperphysics.phy-astr.gsu.edu/hbase/trgser.html#c1)in which case Newton's 2nd law (http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#fma) takes the form
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend11.gifEven in this approximate case, the solution of the equation uses calculus and differential equations. The differential equation (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c5) is
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend12.gifand for small angles θ the solution is:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend13.gifhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend14.gifPendulum Equation
The equation of motion for the simple pendulum (http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1) for sufficiently small amplitude has the form
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend16.gifwhich when put in angular form (http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#rq) becomes
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend17.gifThis differential equation (http://hyperphysics.phy-astr.gsu.edu/hbase/diff2.html#c2) is like that for the simple harmonic oscillator (http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c2) and has the solution:
http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend13.gifhttp://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/pend2.gif